A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V (G) and an edge set E(G). A total signed domination function of G is a function f : V (G) ∪ E(G) → {−1, 1}. The weight of f is w(f) = Σx∈V(G)∪E(G)f(x). For an element x ∈ V (G) ∪ E(G), we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f[x] = \sum\nolimits_{y \in N_T [x]} {f(y)} $$ \end{document}. A total signed domination function of G is a function f : V (G) ∪ E(G) → {−1, 1} such that f[x] ≽ 1 for all x ∈ V (G) ∪ E(G). The total signed domination number γs* (G) of G is the minimum weight of a total signed domination function on G.